3.1.24 \(\int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [24]

Optimal. Leaf size=327 \[ \frac {e x}{2 a}+\frac {b^2 e x}{a^3}+\frac {f x^2}{4 a}+\frac {b^2 f x^2}{2 a^3}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b \sqrt {a^2+b^2} f \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b \sqrt {a^2+b^2} f \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b f \sinh (c+d x)}{a^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d} \]

[Out]

1/2*e*x/a+b^2*e*x/a^3+1/4*f*x^2/a+1/2*b^2*f*x^2/a^3-b*(f*x+e)*cosh(d*x+c)/a^2/d-1/4*f*cosh(d*x+c)^2/a/d^2+b*f*
sinh(d*x+c)/a^2/d^2+1/2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a/d-b*(f*x+e)*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(
a^2+b^2)^(1/2)/a^3/d+b*(f*x+e)*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d-b*f*polylog(2,-a*e
xp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^2+b*f*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b
^2)^(1/2)/a^3/d^2

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Rubi [A]
time = 0.41, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5713, 5698, 3391, 5684, 3377, 2717, 3403, 2296, 2221, 2317, 2438} \begin {gather*} \frac {b^2 e x}{a^3}+\frac {b^2 f x^2}{2 a^3}+\frac {b f \sinh (c+d x)}{a^2 d^2}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {b f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^3 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {e x}{2 a}+\frac {f x^2}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]

[Out]

(e*x)/(2*a) + (b^2*e*x)/a^3 + (f*x^2)/(4*a) + (b^2*f*x^2)/(2*a^3) - (b*(e + f*x)*Cosh[c + d*x])/(a^2*d) - (f*C
osh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^3*
d) + (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3*d) - (b*Sqrt[a^2 + b^2]
*f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*Sqrt[a^2 + b^2]*f*PolyLog[2, -((a*E^(c
+ d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*f*Sinh[c + d*x])/(a^2*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c
+ d*x])/(2*a*d)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5684

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[-a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*
x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 5698

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 5713

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F[c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

Rubi steps

\begin {align*} \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx &=\int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx\\ &=\frac {\int (e+f x) \cosh ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{b+a \sinh (c+d x)} \, dx}{a}\\ &=-\frac {f \cosh ^2(c+d x)}{4 a d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {\int (e+f x) \, dx}{2 a}-\frac {b \int (e+f x) \sinh (c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x) \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e+f x}{b+a \sinh (c+d x)} \, dx}{a^3}\\ &=\frac {e x}{2 a}+\frac {b^2 e x}{a^3}+\frac {f x^2}{4 a}+\frac {b^2 f x^2}{2 a^3}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac {\left (2 b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx}{a^3}+\frac {(b f) \int \cosh (c+d x) \, dx}{a^2 d}\\ &=\frac {e x}{2 a}+\frac {b^2 e x}{a^3}+\frac {f x^2}{4 a}+\frac {b^2 f x^2}{2 a^3}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}+\frac {b f \sinh (c+d x)}{a^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac {\left (2 b \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}+\frac {\left (2 b \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}\\ &=\frac {e x}{2 a}+\frac {b^2 e x}{a^3}+\frac {f x^2}{4 a}+\frac {b^2 f x^2}{2 a^3}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b f \sinh (c+d x)}{a^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {\left (b \sqrt {a^2+b^2} f\right ) \int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}-\frac {\left (b \sqrt {a^2+b^2} f\right ) \int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}\\ &=\frac {e x}{2 a}+\frac {b^2 e x}{a^3}+\frac {f x^2}{4 a}+\frac {b^2 f x^2}{2 a^3}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b f \sinh (c+d x)}{a^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {\left (b \sqrt {a^2+b^2} f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}-\frac {\left (b \sqrt {a^2+b^2} f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}\\ &=\frac {e x}{2 a}+\frac {b^2 e x}{a^3}+\frac {f x^2}{4 a}+\frac {b^2 f x^2}{2 a^3}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b \sqrt {a^2+b^2} f \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b \sqrt {a^2+b^2} f \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b f \sinh (c+d x)}{a^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.31, size = 1581, normalized size = 4.83 \begin {gather*} \frac {\text {csch}(c+d x) (b+a \sinh (c+d x)) \left (2 a^2 e \left (\frac {c}{d}+x-\frac {2 b \text {ArcTan}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}\right )+a^2 f \left (x^2+\frac {2 i b \pi \tanh ^{-1}\left (\frac {-a+b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d^2}+\frac {2 b \left (2 \left (c+i \text {ArcCos}\left (-\frac {i b}{a}\right )\right ) \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )+(-2 i c+\pi -2 i d x) \tanh ^{-1}\left (\frac {(-i a+b) \tan \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )-\left (\text {ArcCos}\left (-\frac {i b}{a}\right )-2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {(a+i b) \left (a-i b+\sqrt {-a^2-b^2}\right ) \left (1+i \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {i b}{a}\right )+2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {i (a+i b) \left (-a+i b+\sqrt {-a^2-b^2}\right ) \left (i+\cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )+\left (\text {ArcCos}\left (-\frac {i b}{a}\right )+2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )-2 i \tanh ^{-1}\left (\frac {(-i a+b) \tan \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} \sqrt {-a^2-b^2} e^{-\frac {c}{2}-\frac {d x}{2}}}{\sqrt {2} \sqrt {-i a} \sqrt {b+a \sinh (c+d x)}}\right )+\left (\text {ArcCos}\left (-\frac {i b}{a}\right )-2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )+2 i \tanh ^{-1}\left (\frac {(-i a+b) \tan \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {-a^2-b^2} e^{\frac {1}{2} (c+d x)}}{\sqrt {2} \sqrt {-i a} \sqrt {b+a \sinh (c+d x)}}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (i b+\sqrt {-a^2-b^2}\right ) \left (a+i b-i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (b+i \sqrt {-a^2-b^2}\right ) \left (i a-b+\sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )\right )\right )}{\sqrt {-a^2-b^2} d^2}\right )+\frac {2 e \left (\left (a^2+4 b^2\right ) (c+d x)-\frac {2 b \left (3 a^2+4 b^2\right ) \text {ArcTan}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \cosh (c+d x)+a^2 \sinh (2 (c+d x))\right )}{d}+\frac {f \left (\left (a^2+4 b^2\right ) (-c+d x) (c+d x)-8 a b d x \cosh (c+d x)-a^2 \cosh (2 (c+d x))-\frac {2 b \left (3 a^2+4 b^2\right ) \left (2 c \tanh ^{-1}\left (\frac {b+a \cosh (c+d x)+a \sinh (c+d x)}{\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {a (\cosh (c+d x)+\sinh (c+d x))}{b-\sqrt {a^2+b^2}}\right )-(c+d x) \log \left (1+\frac {a (\cosh (c+d x)+\sinh (c+d x))}{b+\sqrt {a^2+b^2}}\right )+\text {PolyLog}\left (2,\frac {a (\cosh (c+d x)+\sinh (c+d x))}{-b+\sqrt {a^2+b^2}}\right )-\text {PolyLog}\left (2,-\frac {a (\cosh (c+d x)+\sinh (c+d x))}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+8 a b \sinh (c+d x)+2 a^2 d x \sinh (2 (c+d x))\right )}{d^2}\right )}{8 a^3 (a+b \text {csch}(c+d x))} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]

[Out]

(Csch[c + d*x]*(b + a*Sinh[c + d*x])*(2*a^2*e*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2
]])/(Sqrt[-a^2 - b^2]*d)) + a^2*f*(x^2 + ((2*I)*b*Pi*ArcTanh[(-a + b*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqr
t[a^2 + b^2]*d^2) + (2*b*(2*(c + I*ArcCos[((-I)*b)/a])*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sq
rt[-a^2 - b^2]] + ((-2*I)*c + Pi - (2*I)*d*x)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a
^2 - b^2]] - (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*L
og[((a + I*b)*(a - I*b + Sqrt[-a^2 - b^2])*(1 + I*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^
2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] - (ArcCos[((-I)*b)/a] + 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi +
(2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[(I*(a + I*b)*(-a + I*b + Sqrt[-a^2 - b^2])*(I + Cot[((2*I)*c + Pi + (2*I
)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] + (ArcCos[((-I)*b)/a] + 2*Ar
cTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (2*I)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*
c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[-a^2 - b^2]*E^(-1/2*c - (d*x)/2))/(Sqrt[2]*S
qrt[(-I)*a]*Sqrt[b + a*Sinh[c + d*x]]))] + (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)
*d*x)/4])/Sqrt[-a^2 - b^2]] + (2*I)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]]
)*Log[((-1)^(1/4)*Sqrt[-a^2 - b^2]*E^((c + d*x)/2))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a*Sinh[c + d*x]])] + I*(Pol
yLog[2, ((I*b + Sqrt[-a^2 - b^2])*(a + I*b - I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*
b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] - PolyLog[2, ((b + I*Sqrt[-a^2 - b^2])*(I*a - b +
Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I
)*d*x)/4]))])))/(Sqrt[-a^2 - b^2]*d^2)) + (2*e*((a^2 + 4*b^2)*(c + d*x) - (2*b*(3*a^2 + 4*b^2)*ArcTan[(a - b*T
anh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + a^2*Sinh[2*(c + d*x)]))/d + (f*(
(a^2 + 4*b^2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - a^2*Cosh[2*(c + d*x)] - (2*b*(3*a^2 + 4*b^2)*(2
*c*ArcTanh[(b + a*Cosh[c + d*x] + a*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (a*(Cosh[c + d*x] + Si
nh[c + d*x]))/(b - Sqrt[a^2 + b^2])] - (c + d*x)*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b
^2])] + PolyLog[2, (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])] - PolyLog[2, -((a*(Cosh[c + d*x
] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*a^2*d*x*Sinh[2*(c + d*
x)]))/d^2))/(8*a^3*(a + b*Csch[c + d*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1011\) vs. \(2(297)=594\).
time = 5.94, size = 1012, normalized size = 3.09

method	result	size
risch	\(\frac {f \,x^{2}}{4 a}+\frac {e x}{2 a}+\frac {b^{2} f \,x^{2}}{2 a^{3}}+\frac {b^{2} e x}{a^{3}}+\frac {\left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{2}}-\frac {b \left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 a^{2} d^{2}}-\frac {b \left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 a^{2} d^{2}}-\frac {\left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{2}}+\frac {2 b e \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d a \sqrt {a^{2}+b^{2}}}+\frac {2 b^{3} e \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {b f \dilog \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \dilog \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \dilog \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \dilog \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}-\frac {2 b f c \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {2 b^{3} f c \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}\)	\(1012\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/4*f*x^2/a+1/2*e*x/a+1/2*b^2*f*x^2/a^3+b^2*e*x/a^3+1/16*(2*d*f*x+2*d*e-f)/a/d^2*exp(2*d*x+2*c)-1/2*b*(d*f*x+d
*e-f)/a^2/d^2*exp(d*x+c)-1/2*b*(d*f*x+d*e+f)/a^2/d^2*exp(-d*x-c)-1/16*(2*d*f*x+2*d*e+f)/a/d^2*exp(-2*d*x-2*c)+
2/d/a*b*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))+2/d/a^3*b^3*e/(a^2+b^2)^(1/2)*arct
anh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))-1/d/a*b*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(
-b+(a^2+b^2)^(1/2)))*x-1/d^2/a*b*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*
c+1/d/a*b*f/(a^2+b^2)^(1/2)*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*x+1/d^2/a*b*f/(a^2+b^2)^(
1/2)*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*c-1/d^2/a*b*f/(a^2+b^2)^(1/2)*dilog((-a*exp(d*x+
c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))+1/d^2/a*b*f/(a^2+b^2)^(1/2)*dilog((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)
/(b+(a^2+b^2)^(1/2)))-1/d/a^3*b^3*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))
*x-1/d^2/a^3*b^3*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*c+1/d/a^3*b^3*f/
(a^2+b^2)^(1/2)*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*x+1/d^2/a^3*b^3*f/(a^2+b^2)^(1/2)*ln(
(a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*c-1/d^2/a^3*b^3*f/(a^2+b^2)^(1/2)*dilog((-a*exp(d*x+c)+(
a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))+1/d^2/a^3*b^3*f/(a^2+b^2)^(1/2)*dilog((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)
/(b+(a^2+b^2)^(1/2)))-2/d^2/a*b*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))-2/d^2/a^
3*b^3*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="maxima")

[Out]

-1/16*(32*(a^2*b*e^c + b^3*e^c)*integrate(x*e^(d*x)/(a^4*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^4), x) - (4
*(a^2*d^2*e^(2*c) + 2*b^2*d^2*e^(2*c))*x^2 + (2*a^2*d*x*e^(4*c) - a^2*e^(4*c))*e^(2*d*x) - 8*(a*b*d*x*e^(3*c)
- a*b*e^(3*c))*e^(d*x) - 8*(a*b*d*x*e^c + a*b*e^c)*e^(-d*x) - (2*a^2*d*x + a^2)*e^(-2*d*x))*e^(-2*c)/(a^3*d^2)
)*f - 1/8*((4*b*e^(-d*x - c) - a)*e^(2*d*x + 2*c)/(a^2*d) - 4*(a^2 + 2*b^2)*(d*x + c)/(a^3*d) + (4*b*e^(-d*x -
c) + a*e^(-2*d*x - 2*c))/(a^2*d) + 8*(a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c)
- b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d))*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1484 vs. \(2 (301) = 602\).
time = 0.45, size = 1484, normalized size = 4.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="fricas")

[Out]

-1/16*(2*a^2*d*f*x - (2*a^2*d*f*x + 2*a^2*d*cosh(1) + 2*a^2*d*sinh(1) - a^2*f)*cosh(d*x + c)^4 - (2*a^2*d*f*x
+ 2*a^2*d*cosh(1) + 2*a^2*d*sinh(1) - a^2*f)*sinh(d*x + c)^4 + 2*a^2*d*cosh(1) + 8*(a*b*d*f*x + a*b*d*cosh(1)
+ a*b*d*sinh(1) - a*b*f)*cosh(d*x + c)^3 + 2*a^2*d*sinh(1) + 4*(2*a*b*d*f*x + 2*a*b*d*cosh(1) + 2*a*b*d*sinh(1
) - 2*a*b*f - (2*a^2*d*f*x + 2*a^2*d*cosh(1) + 2*a^2*d*sinh(1) - a^2*f)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2*f
- 4*((a^2 + 2*b^2)*d^2*f*x^2 + 2*(a^2 + 2*b^2)*d^2*x*cosh(1) + 2*(a^2 + 2*b^2)*d^2*x*sinh(1))*cosh(d*x + c)^2
- 2*(2*(a^2 + 2*b^2)*d^2*f*x^2 + 4*(a^2 + 2*b^2)*d^2*x*cosh(1) + 4*(a^2 + 2*b^2)*d^2*x*sinh(1) + 3*(2*a^2*d*f
*x + 2*a^2*d*cosh(1) + 2*a^2*d*sinh(1) - a^2*f)*cosh(d*x + c)^2 - 12*(a*b*d*f*x + a*b*d*cosh(1) + a*b*d*sinh(1
) - a*b*f)*cosh(d*x + c))*sinh(d*x + c)^2 + 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d*x + c)*sinh(d*x + c) +
a*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*s
inh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) - 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d*x + c)*sinh(d*x +
c) + a*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c)
+ a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 16*((a*b*c*f - a*b*d*cosh(1) - a*b*d*sinh(1))*cosh(d*x
+ c)^2 + 2*(a*b*c*f - a*b*d*cosh(1) - a*b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a*b*c*f - a*b*d*cosh(1) -
a*b*d*sinh(1))*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a
^2 + b^2)/a^2) + 2*b) - 16*((a*b*c*f - a*b*d*cosh(1) - a*b*d*sinh(1))*cosh(d*x + c)^2 + 2*(a*b*c*f - a*b*d*cos
h(1) - a*b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a*b*c*f - a*b*d*cosh(1) - a*b*d*sinh(1))*sinh(d*x + c)^2)
*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 16*((a*b
*d*f*x + a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*f*x + a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f*x + a*b*c*f
)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d
*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) - 16*((a*b*d*f*x + a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*f*x + a*b*c*f)*c
osh(d*x + c)*sinh(d*x + c) + (a*b*d*f*x + a*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x + c
) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) + 8*(a*b*d*f*x + a*b*d
*cosh(1) + a*b*d*sinh(1) + a*b*f)*cosh(d*x + c) + 4*(2*a*b*d*f*x + 2*a*b*d*cosh(1) - (2*a^2*d*f*x + 2*a^2*d*co
sh(1) + 2*a^2*d*sinh(1) - a^2*f)*cosh(d*x + c)^3 + 2*a*b*d*sinh(1) + 2*a*b*f + 6*(a*b*d*f*x + a*b*d*cosh(1) +
a*b*d*sinh(1) - a*b*f)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d^2*f*x^2 + 2*(a^2 + 2*b^2)*d^2*x*cosh(1) + 2*(a^2 +
2*b^2)*d^2*x*sinh(1))*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^2*cosh(d*x + c)^2 + 2*a^3*d^2*cosh(d*x + c)*sinh(d
*x + c) + a^3*d^2*sinh(d*x + c)^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right ) \cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)**2/(a+b*csch(d*x+c)),x)

[Out]

Integral((e + f*x)*cosh(c + d*x)**2/(a + b*csch(c + d*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)*cosh(d*x + c)^2/(b*csch(d*x + c) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)^2*(e + f*x))/(a + b/sinh(c + d*x)),x)

[Out]

int((cosh(c + d*x)^2*(e + f*x))/(a + b/sinh(c + d*x)), x)

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